Solve The System Of Linear Equations By Graphing.$\[ \begin{array}{l} y = -\frac{5}{2}x - 7 \\ x + 2y = 4 \end{array} \\]What Is The Solution To The System Of Linear Equations?A. \[$(-4.5, 4.25)\$\] B. \[$(-1.7, -2.8)\$\] C.

Introduction

In this article, we will explore the method of solving a system of linear equations by graphing. This method involves graphing the two linear equations on the same coordinate plane and finding the point of intersection, which represents the solution to the system. We will use the given system of linear equations:

{ \begin{array}{l} y = -\frac{5}{2}x - 7 \\ x + 2y = 4 \end{array} \}

Graphing the First Equation

The first equation is already in slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this case, the slope is $-\frac{5}{2}$ and the y-intercept is $-7$. To graph this equation, we can use the slope and y-intercept to find two points on the line.

Finding Two Points on the Line

To find the first point, we can plug in $x = 0$ into the equation:

$y = -\frac{5}{2}(0) - 7$ $y = -7$

So, the first point is $(0, -7)$.

To find the second point, we can plug in $x = 2$ into the equation:

$y = -\frac{5}{2}(2) - 7$ $y = -10$

So, the second point is $(2, -10)$.

Graphing the Second Equation

The second equation is not in slope-intercept form, so we need to solve it for $y$ to graph it. We can use the method of substitution or elimination to solve for $y$.

Solving for $y$

Let's use the method of substitution to solve for $y$. We can solve the second equation for $x$:

$x = 4 - 2y$

Now, we can substitute this expression for $x$ into the first equation:

$y = -\frac{5}{2}(4 - 2y) - 7$

Expanding and simplifying, we get:

$y = -10 + 5y - 7$ $y = -17 + 5y$

Subtracting $5y$ from both sides, we get:

$-4y = -17$ $y = \frac{17}{4}$

Now, we can substitute this expression for $y$ into the second equation to solve for $x$:

$x + 2(\frac{17}{4}) = 4$

Simplifying, we get:

$x + \frac{17}{2} = 4$ $x = 4 - \frac{17}{2}$ $x = -\frac{11}{2}$

So, the point of intersection is $(-\frac{11}{2}, \frac{17}{4})$.

Finding the Solution

To find the solution to the system of linear equations, we need to find the point of intersection between the two lines. We can do this by graphing the two lines on the same coordinate plane and finding the point where they intersect.

Graphing the Lines

To graph the lines, we can use the points we found earlier. The first line passes through the points $(0, -7)$ and $(2, -10)$. The second line passes through the point $(-\frac{11}{2}, \frac{17}{4})$.

Finding the Point of Intersection

To find the point of intersection, we can use the fact that the two lines intersect at a single point. We can find this point by finding the point where the two lines are equal.

Solving for $x$

We can set the two equations equal to each other and solve for $x$:

$-\frac{5}{2}x - 7 = x + 2(\frac{17}{4})$

Simplifying, we get:

$-\frac{5}{2}x - 7 = x + \frac{17}{2}$

Multiplying both sides by $2$, we get:

$-5x - 14 = 2x + 17$

Adding $5x$ to both sides, we get:

$-14 = 7x + 17$

Subtracting $17$ from both sides, we get:

$-31 = 7x$

Dividing both sides by $7$, we get:

$x = -\frac{31}{7}$

Solving for $y$

Now that we have found the value of $x$, we can substitute it into one of the original equations to solve for $y$. Let's use the first equation:

$y = -\frac{5}{2}(-\frac{31}{7}) - 7$

Simplifying, we get:

$y = \frac{155}{14} - 7$ $y = \frac{155 - 98}{14}$ $y = \frac{57}{14}$

So, the solution to the system of linear equations is $(-\frac{31}{7}, \frac{57}{14})$.

Conclusion

In this article, we used the method of graphing to solve a system of linear equations. We graphed the two lines on the same coordinate plane and found the point of intersection, which represents the solution to the system. We also used the method of substitution to solve for $y$ and found the value of $x$ by setting the two equations equal to each other. The solution to the system of linear equations is $(-\frac{31}{7}, \frac{57}{14})$.

Answer

The correct answer is:

A. $(-4.5, 4.25)$ is not the correct solution.

B. $(-1.7, -2.8)$ is not the correct solution.

C. $(-\frac{31}{7}, \frac{57}{14})$ is the correct solution.

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Introduction

In our previous article, we explored the method of solving a system of linear equations by graphing. We graphed the two lines on the same coordinate plane and found the point of intersection, which represents the solution to the system. In this article, we will answer some common questions related to solving systems of linear equations by graphing.

Q: What is the main advantage of solving systems of linear equations by graphing?

A: The main advantage of solving systems of linear equations by graphing is that it allows us to visualize the relationship between the two equations and find the point of intersection, which represents the solution to the system.

Q: What are some common mistakes to avoid when graphing systems of linear equations?

A: Some common mistakes to avoid when graphing systems of linear equations include:

• Not using a ruler or straightedge to draw the lines accurately
• Not labeling the axes and the point of intersection
• Not checking for any errors in the graphing process
• Not using a calculator or computer to check the solution

Q: How can I determine if the point of intersection is the only solution to the system?

A: To determine if the point of intersection is the only solution to the system, you can check if the two lines intersect at a single point. If the lines intersect at a single point, then the point of intersection is the only solution to the system.

Q: What if the two lines do not intersect? What does this mean?

A: If the two lines do not intersect, then the system of linear equations has no solution. This means that the two equations are inconsistent and cannot be true at the same time.

Q: Can I use graphing to solve systems of linear equations with more than two variables?

A: No, graphing is not suitable for solving systems of linear equations with more than two variables. Graphing is only suitable for solving systems of linear equations with two variables.

Q: How can I check my solution to a system of linear equations using graphing?

A: To check your solution to a system of linear equations using graphing, you can graph the two lines on the same coordinate plane and check if the point of intersection is the same as the solution you found. If the point of intersection is the same as the solution you found, then your solution is correct.

Q: What are some real-world applications of solving systems of linear equations by graphing?

A: Some real-world applications of solving systems of linear equations by graphing include:

• Finding the intersection of two roads or highways
• Determining the point of intersection of two lines in a coordinate plane
• Finding the solution to a system of linear equations in a business or economics problem

Conclusion

In this article, we answered some common questions related to solving systems of linear equations by graphing. We discussed the main advantage of solving systems of linear equations by graphing, common mistakes to avoid, and how to determine if the point of intersection is the only solution to the system. We also discussed real-world applications of solving systems of linear equations by graphing.

Additional Resources

For more information on solving systems of linear equations by graphing, you can check out the following resources:

• Khan Academy: Solving Systems of Linear Equations by Graphing
• Mathway: Solving Systems of Linear Equations by Graphing
• Wolfram Alpha: Solving Systems of Linear Equations by Graphing

Note: The resources listed above are just a few examples of the many resources available online for solving systems of linear equations by graphing.